30-60-90 Right Triangles
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: May 22, 2016
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
www.ck12.org
C HAPTER
Chapter 1. 30-60-90 Right Triangles
1
30-60-90 Right Triangles
Here you’ll learn that the sides of a 30-60-90 right triangle are in the ratio x : x
√
3 : 2x.
What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the
lengths of its other sides? After completing this Concept, you’ll be able to use the 30-60-90 Theorem to solve
problems like this one.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136654
CK-12 Foundation: Special Right Triangle: 30-60-90
Watch the first half of this video.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/1362
James Sousa: Trigonometric Function Values of Special Angles
Now watch the first half of this video.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/1363
James Sousa: Solving Special Right Triangles
Guidance
One of the two special right triangles is called a 30-60-90 triangle, after its three angles.
√
30-60-90 Theorem: If a triangle has angle measures 30◦ , 60◦ and 90◦ , then the sides are in the ratio x : x 3 : 2x.
√
The shorter leg is always x, the longer leg is always x 3, and the hypotenuse is always 2x. If you ever forget these
theorems, you can still use the Pythagorean Theorem.
1
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Example A
Find the length of the missing side.
√
We are given the shorter leg. If x = 5, then the longer leg, b = 5 3, and the hypotenuse, c = 2(5) = 10.
Example B
Find the length of the missing side.
We are given the hypotenuse. 2x = 20, so the shorter leg, f =
20
2
√
= 10, and the longer leg, g = 10 3.
Example C
√
A rectangle has sides 4 and 4 3. What is the length of the diagonal?
If you are not given a picture, draw one.
The two lengths are x, x
√
3, so the diagonal would be 2x, or 2(4) = 8.
If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.
√ 2
42 + 4 3 = d 2
16 + 48 = d 2
√
d = 64 = 8
2
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Chapter 1. 30-60-90 Right Triangles
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/136655
CK-12 Foundation: Special Right Triangle: 30-60-90
–>
Guided Practice
Find the value of x and y.
1.
2.
3. x is the hypotenuse of a 30-60-90 triangle and y is the longer leg of the same triangle. The shorter leg has a length
of 6.
Answers:
1. We are given the longer leg.
x
√
3 = 12
√
√
√
12
3 12 3
x= √ · √ =
=4 3
3
3
3
The hypotenuse is
√
√
y = 2(4 3) = 8 3
2. We are given the hypotenuse.
2x = 16
x=8
The longer leg is
√
√
y = 8· 3 = 8 3
3
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3. We are given the shorter leg.
x = 2(6)
x = 12
The longer leg is
√
√
y = 6· 3 = 6 3
Explore More
1. In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
2. In a 30-60-90 triangle, if the shorter leg is x, then the longer leg is __________ and the hypotenuse is ___________.
√
3. A rectangle has sides of length 6 and 6 3. What is the length of the diagonal?
4. Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides?
For questions 5-12, find the lengths of the missing sides. Simplify all radicals.
5.
6.
7.
8.
9.
4
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Chapter 1. 30-60-90 Right Triangles
10.
11.
12.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 8.6.
5